The visitors launched a comeback in the second half as Simy prodded in his second before Junior Messias curled in a stunning equaliser in the 59th minute.
Nigeria international Simy's double continued a superb run of form that makes him the top scorer in Europe's top-five leagues since the start of March with eight goals.
But the relegation battlers could not hold on as Giovanni Di Lorenzo rifled in a shot to put Gennaro Gattuso's side back in front after 72 minutes.
Napoli climbed one point above Juventus, who face Torino later on Saturday, into fourth place on 56 points, but they remained two points behind Atalanta in third after the Bergamo club beat Udinese 3-2.
Manuel Lazzari's opener was cancelled out by a spectacular bicycle kick from Spezia substitute Daniele Verde before the late spot-kick, but Lazzari and Joaquin Correa were sent off for the home side in stoppage time.
Lorenzo Pellegrini's penalty edged Roma ahead at halftime in an entertaining game.
Elsewhere, relegation battlers Benevento and Parma played out a 2-2 draw, Hellas Verona went seventh with a 2-0 win at Cagliari and Franck Ribery was sent off during Fiorentina's 1-1 draw at Genoa on Giuseppe Iachini's return as Viola coach.